Example 1

Step 1: Determine the conjugate of the denominator.

The conjugate of the denominator is \(\sqrt 3 + \sqrt 6 \).

Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, and then simplify.

\(\begin{align}
 \frac{3}{{\sqrt 3 - \sqrt 6 }} &= \frac{3}{{\left( {\sqrt 3 - \sqrt 6 } \right)}}\cdot \frac{{\left( {\sqrt 3 + \sqrt 6 } \right)}}{{\left( {\sqrt 3 + \sqrt 6 } \right)}} \\ 
  &= \frac{{3\left( {\sqrt 3 + \sqrt 6 } \right)}}{{\left( {\sqrt 3 - \sqrt 6 } \right)\left( {\sqrt 3 + \sqrt 6 } \right)}} \\ 
  &= \frac{{3\left( {\sqrt 3 + \sqrt 6 } \right)}}{{\sqrt {3^2 } {\cancel{ +\sqrt 3 \cdot \sqrt 6}}{\cancel {-\sqrt 3 \cdot \sqrt 6}} - \sqrt {6^2 } }} \\ 
  &= \frac{{3\left( {\sqrt 3 + \sqrt 6 } \right)}}{{\sqrt {3^2 } - \sqrt {6^2 } }} \\ 
  &= \frac{{3\left( {\sqrt 3 + \sqrt 6 } \right)}}{{3 - 6}} \\ 
  &= \frac{{3\left( {\sqrt 3 + \sqrt 6 } \right)}}{{ - 3}} \\ 
  &= - \left( {\sqrt 3 + \sqrt 6 } \right) \\ 
  &= - \sqrt 3 - \sqrt 6  \\ 
 \end{align}\)